cp's OEIS Frontend

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

%I A242671 #52 Mar 19 2025 08:36:49
%S A242671 7,2,3,6,0,6,7,9,7,7,4,9,9,7,8,9,6,9,6,4,0,9,1,7,3,6,6,8,7,3,1,2,7,6,
%T A242671 2,3,5,4,4,0,6,1,8,3,5,9,6,1,1,5,2,5,7,2,4,2,7,0,8,9,7,2,4,5,4,1,0,5,
%U A242671 2,0,9,2,5,6,3,7,8,0,4,8,9,9,4,1,4,4,1,4,4,0,8,3,7,8,7,8,2,2,7,4
%N A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.
%C A242671 Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
%C A242671 Essentially the same as A229780, A134972, A134945, A098317 and A002163. - _R. J. Mathar_, May 23 2014
%C A242671 Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - _Geoffrey Critzer_, Feb 04 2022
%C A242671 The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - _Amiram Eldar_, Mar 18 2025
%D A242671 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.
%H A242671 Alfréd Rényi, <a href="https://static.renyi.hu/renyi_cikkek/1957_representations_for_real_numbers_and_their_ergodic_properties.pdf">Representations for real numbers and their ergodic properties</a>, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
%H A242671 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A242671 Equals (1 + 1/sqrt(5))/2.
%F A242671 Equals 1/A094874. - _Michel Marcus_, Dec 01 2018
%F A242671 From _Amiram Eldar_, Feb 11 2022: (Start)
%F A242671 Equals phi/sqrt(5), where phi is the golden ratio (A001622).
%F A242671 Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
%F A242671 From _Amiram Eldar_, Nov 28 2024: (Start)
%F A242671 Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
%F A242671 Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
%F A242671 Equals 1 - A244847. - _Amiram Eldar_, Mar 18 2025
%e A242671 k2 = 0.723606797749978969640917366873127623544...
%t A242671 RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First
%o A242671 (PARI) (1 + 1/sqrt(5))/2 \\ _Stefano Spezia_, Dec 07 2024
%Y A242671 Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.
%K A242671 nonn,cons,easy
%O A242671 0,1
%A A242671 _Jean-François Alcover_, May 20 2014